Laplacian Eigenmaps and Bayesian Clustering Based Layout Pattern - - PowerPoint PPT Presentation

Laplacian Eigenmaps and Bayesian Clustering Based Layout Pattern Sampling and Its Applications to Hotspot Detection and OPC

Tetsuaki Matsunawa1, Bei Yu2 and David Z. Pan3

1Toshiba Corporation 2The Chinese University of Hong Kong 3The University of Texas at Austin

2

Outline

• Background
• Pattern Sampling in Physical Verification
• Overall flow
• Laplacian Eigenmaps
• Bayesian Clustering
• Applications

– Lithography Hotspot Detection – OPC (Optical Proximity Correction)

• Conclusion

3

Background

• Issue: Systematic method for pattern sampling is not established
• Goal: Pattern sampling automation for process optimization

A A A A A B B B B B C C C C C A B C

Grouping

Representative patterns

1D patterns 2D patterns Based on engineer’s knowledge

Test patterns for :

Simulation model calibration Source mask optimization Wafer verification, etc.

4

Pattern Sampling

Input Layout

x1 = (0, 1, 0, 1.5, …) x2 = (2, 0.5, 1, -1, …) x3 = (1, -1, 0, 0.3, …) ……

Feature extraction

dimension 1 dimension 2 dimension 1 dimension 2

Dimension Reduction Sampling

5

Pattern Sampling in Physical Verification

• Key techniques: Dimension reduction and Clustering

I.

• W. C. Tam, et al., “Systematic Defect Identification through

Layout Snippet Clustering,” ITC, 2010 II.

• S. Shim, et al., “Synthesis of Lithography Test Patterns

through Topology-Oriented Pattern Extraction and Classification,” SPIE, 2014

• III. V. Dai, et al., “Systematic Physical Verification with

Topological Patterns,” SPIE, 2014

group group [I] W. C. Tam [II] S. Shim Examples of clustering results Classification flow

6

Open Questions

• Undefined similarity
• A criterion for defining pattern similarity to

evaluate essential characteristics in real layouts is unclear

• Manual parameter tuning
• Most clustering algorithms require several

preliminary experiments (total number of clusters)

7

Laplacian Eigenmaps and Bayesian Clustering

• We develop

– An efficient feature comparison method

• With nonlinear dimensionality reduction / kernel

parameter optimization

– An automated pattern sampling using Bayesian model based clustering

• Without manual parameter tuning

8

• Problem: Given layout data, a classification model is

trained to extract representative patterns

• Goal: To classify the layout patterns into a set of classes

minimizing the Bayes error

Problem formulation: Layout Pattern Sampling

𝒛 = 𝒈 𝒚

Unique pattern set

Pattern ID Frequency

…

Classification model Output (y) Input (x)

Layout data

9

Bayes Error (BE)

𝐶𝐹 = ' min 1 − 𝑞 𝜕/|𝑦 𝑞 𝑦 𝑒𝑦

Bayes Error = 0.02, WCS/BCS = 0.07 Bayes Error = 1.68, WCS/BCS = 0.07 Comparison between BE and Within-Class Scatter/Between-Class Scatter

• To quantify the clustering performance

– Define a quality of clustering distributions based on Bayes’ theorem

𝑄 𝜕|𝑦 : conditional probability in class 𝜕 𝑄 𝑦 : prior probability

• f data 𝑦

10

Overall Flow

(1) Sampling phase (2) Application phase

Model training for lHotspot detection, lMask Optimization, lProcess Simulation, lWafer Inspection, etc.

Sample Plan Application

GDSII Layout

Feature A Feature B Feature C

⁞

Low-dimensional vectors A

⁞

Layout Feature Extraction Dimensionality Reduction Clustering

Layout dataset A Low-dimensional vectors B Low-dimensional vectors C Layout dataset B Layout dataset C

⁞

Locating Feature Points

DRC

Ranked dataset

(Feature A, B or C)

Ranking

Ranked dataset

(Feature A, B or C)

Ranked dataset

(Feature A, B or C)

⁞

11

Feature Point Generation & Feature Extraction

0.0 0.3 0.0 0.3 0.0 0.0 0.3 0.0 0.3 0.0 0.0 0.4 0.3 0.4 0.0 0.0 0.3 0.0 0.3 0.0 0.0 0.3 0.0 0.3 0.0 Feature point Unique pattern TLineEnd l

GDS Locating feature points Feature extraction

Density based encoding Diffraction order distribution

12

Why dimension reduction and Bayesian clustering?

Feature A Feature B Feature C

Comparable data Bayes Model

High dimension Low dimension

Dimension Reduction Required feature comparison for optimal feature selection

Ø The optimal characteristics for layout representation vary in different applications

How to compare diverse layout feature types?

Ø #of dimensions differs with different types of features

Hard to achieve completely automatic clustering

Ø Hypothetical parameters are required for typical clustering task

Automatic Clustering

13

Laplacian Eigenmaps

Original data (3D) Linear(2D)

Principal Component Analysis

Nonlinear(2D)

Laplacian Eigenmaps

Comparison with linear/nonlinear algorithm 𝑀𝜔 = 𝛿𝐸𝜔

𝐸 = diag < 𝑋>,>@

A >@BC

𝑋>,>@ = D 1 if 𝑦> ∈𝑙𝑂𝑂 𝑦>@ 𝑝𝑠 𝑦>@ ∈ 𝑙𝑂𝑂 𝑦>

• therwise

𝑀 = 𝐸 − 𝑋

Solve an eigenvalue problem:

Laplacian matrix Diagonal matrix Kernel : k-nearest neighbors

To reduce dimensions while preserving complicated structure

14

Kernel Parameter Optimization

• Optimization through estimating density-ratio 𝑠̂ 𝐲 = 𝐱𝚾 𝐲

between given feature vectors 𝑄 𝑦 and embedded feature vectors 𝑄′ 𝑦 max

Y ∑

log 𝑥]𝜚 𝑦>

_ A_ >BC

Subject to ∑ 𝑥]𝜚 𝑦> = 𝑜 and 𝑥 ≥ 0

A >BC

This is convex optimization, so repeating gradient ascent and constraint satisfaction converges to global solution

𝑠̂ 𝑦 𝑠 𝑦 = 𝑄′ 𝑦 𝑄 𝑦 𝑄′ 𝑦 𝑄 𝑦 𝑜′: #of test samples 𝑜 : #of training samples

15

Bayesian Clustering

Data k1 k2 k3

…

4 𝛽 + 𝑜 − 1 2 𝛽 + 𝑜 − 1 𝛽 𝛽 + 𝑜 − 1

Clusters Prior probability :

x1 x2 x3 x4 x5 x6 xn … x1

𝑞 𝐲|𝛽,𝑞 𝛊 = < 𝜌/𝒪 𝜈/,𝜏/

k /BC

mixture ratio Gaussian distribution

𝑞 𝑨A = 𝑙|𝑦A,𝑨C,…, 𝑨AnC ∝ 𝑞 𝑦A|𝑙 𝑜/ 𝛽 + 𝑜 − 1 𝑙 = 1 ⋯𝐿 𝑞 𝑦A|𝑙ArY 𝛽 𝛽 + 𝑜 − 1 𝑙 = 𝐿 + 1

Centroid Similarity

• Clustering automation without arbitrary parameter tuning
• Bayesian based method: express a parameter distribution

as an infinite dimensional distribution

16

Experiments

• Pattern sampling

– Comparison of conventional methods

• Dimensionality reduction

– Principal Component Analysis (PCA) vs. Laplacian Eigenmaps (LE)

• Clustering

– K-means (Km) vs. Bayesian clustering (BC)

• Applications to

– Lithography Hotspot Detection – OPC

17

Effectiveness of Pattern Sampling

• Representative patterns could be automatically selected

Clustering results: #of extracted patterns Misclassification error rate: Bayes Error

■PCA+Km ■LE+Km ■PCA+BC ■LE+BC #of extracted patterns BE ■PCA+Km ■LE+Km ■PCA+BC ■LE+BC Ratio: ■PCA+Km: 1.0 ■LE+Km: 5.6 ■PCA+BC: 0.7 ■LE+BC: 0.5

18

Application to Lithography Hotspot Detection

• To detect hotspot in short runtime
• Experiments

– Detection model training with different patterns

• PCA+Km, LE+Km, PCA+BC, LE+BC
• Learning algorithm is fixed to Adaptive Boosting

(AdaBoost) – Metrics: detection accuracy and false alarm

19

Effectiveness of Hotspot Detection

• Comparison with conventional clustering method
• Result: Proposed framework achieved the best false-alarm

Detection accuracy: #correctly detected hotspots / #total hotspots False alarm: #correctly detected hotspots / #falsely detected hotspots

■PCA+Km ■LE+Km ■PCA+BC ■LE+BC ■PCA+Km ■LE+Km ■PCA+BC ■LE+BC

20

Application to Regression-based OPC

• To predict edge movements in short runtime
• Experiments

– Prediction model training with different patterns

• PCA+Km, LE+Km, PCA+BC, LE+BC
• Learning algorithm is fixed to Linear regression

– Metric: RMSPE (Root Mean Square Prediction Error)

Iteration : 0 Iteration : 5 Printed image Mask image Predicted edge movements

Regression based method Conventional model-based OPC

(time consuming)

21

Effectiveness of OPC regression

• Proposed framework achieved the best prediction accuracy

■PCA+Km ■LE+Km ■PCA+BC ■LE+BC RMSPE(nm) Ratio: ■PCA+Km: 1.0 ■LE+Km: 1.1 ■PCA+BC: 0.9 ■LE+BC: 0.8

Prediction accuracy: RMSPE: Root mean square prediction error

22

Conclusion ØWe have introduced a new method to sample unique patterns.

ØBy applying our dimension reduction technique, dimensionality- and type-independent layout feature can be used in accordance with applications. ØThe Bayesian clustering is able to classify layout data without manual parameter tuning. ØThe experimental results show that our proposed method can effectively sample layout patterns that represent characteristics of whole chip layout.